Problem 71 - Algebra A Combined Function

Question

Factor. $$ 8 x^{3}+27 y^{3} $$

Step-by-Step Solution

Answer

The expression factors to \((2x + 3y)(4x^2 - 6xy + 9y^2)\).

Step 1: Identify the Sum of Cubes Formula

The expression is a sum of cubes. Recall the formula for factoring a sum of cubes: \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\] We will use this formula to factor the expression \(8x^3 + 27y^3\).

Step 2: Identify 'a' and 'b'

Identify the values of \(a\) and \(b\) that make up the cubes in the expression:\(a = 2x\) because \((2x)^3 = 8x^3\) and \(b = 3y\) because \((3y)^3 = 27y^3\).

Step 3: Substitute into the Formula

Substitute \(a\) and \(b\) into the sum of cubes formula:\[(2x + 3y)((2x)^2 - (2x)(3y) + (3y)^2)\]

Step 4: Simplify the Second Term

Carry out the multiplication and simplifications in the second term:\((2x)^2 = 4x^2,\)\(-(2x)(3y) = -6xy,\)\((3y)^2 = 9y^2.\)This gives: \[4x^2 - 6xy + 9y^2\]

Step 5: Combine into Final Factored Form

Combine the results to express the completely factored form:\[(2x + 3y)(4x^2 - 6xy + 9y^2)\]

Key Concepts

Sum of Cubes

The sum of cubes is a helpful factoring technique in algebra, particularly when working with expressions of the form \(a^3 + b^3\). This formula allows you to break down polynomials into simpler components, making it easier to manage and solve equations. In the case of \(a^3 + b^3\), the sum of cubes can always be factored as \((a + b)(a^2 - ab + b^2)\).
When applying this technique to a problem, the first step is identifying the cube roots of each term in the original expression. For example, in the expression \(8x^3 + 27y^3\), we recognize that \((2x)^3 = 8x^3\) and \((3y)^3 = 27y^3\). These identifications set \(a = 2x\) and \(b = 3y\). This enables us to substitute these values directly into the sum of cubes formula to start the factoring process.
Understanding the sum of cubes is crucial for simplifying polynomial expressions and solving algebraic equations efficiently.

Factoring Techniques

Factoring techniques are essential tools in algebra, primarily for transforming complex expressions into simpler, more manageable forms. These methods help in solving equations, simplifying expressions, and finding roots of polynomials. The sum of cubes is one of these useful techniques that simplify expressions by applying a specific factorization formula.
When dealing with polynomials, follow these general steps to apply the right factoring technique:

• Identify the form of the polynomial (sum/difference of cubes, quadratic, etc.).
• Choose the appropriate factoring formula (for a sum of cubes, use \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)).
• Substitute the variables into the chosen formula.
• Simplify the resulting expression to find the factored form.

In our specific example, we identify \(8x^3 + 27y^3\) as a sum of cubes and use the corresponding technique to simplify it into \((2x + 3y)(4x^2 - 6xy + 9y^2)\). This approach underscores how factoring transforms polynomials to ease further algebraic procedures.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. These expressions form the backbone of algebra, representing real-world phenomena or abstract quantities.
When working with algebraic expressions, understanding how to manipulate and simplify them is key. This process often involves operations such as:

• Addition and subtraction to combine like terms.
• Multiplication and division to apply distributive properties.
• Factoring to reduce complexity or solve equations.

In our case, the expression \(8x^3 + 27y^3\) is composed of two terms, each raised to the third power. Recognizing it as a sum of cubes allows us to factor it effectively. By simplifying it into \((2x + 3y)(4x^2 - 6xy + 9y^2)\), we can facilitate further algebraic manipulations. This ability to simplify underscores the importance of mastering algebraic expressions to handle more complex mathematical challenges.